Question

For the following G.P.'s, find S_n: p, q, `"q"^2/"p", "q"^3/"p"^2`, ...

Answer 1

p, q, `"q"^2/"p", "q"^3/"p"^2`, ...

Here, a = p, r = `"q"/"p"`

Let `"q"/"p" < 1`

S_n = `("a"(1 - "r"^"n"))/(1 - "r")`, for r < 1

∴ S_n = `("p"[1 - ("q"/"p")^"n"])/(1 - "q"/"p")`

∴ S_n = `"p"^2/"p - q" [1 - ("q"/"p")^"n"]`

Let `"q"/"p" > 1`

S_n = `("a"("r"^"n" - 1))/("r"   1)`, for r > 1

∴ S_n = `("p"[("q"/"p")^"n" - 1])/("q"/"p" - 1)`

= `"p"^2/"q - p"[("q"/"p")^"n" - 1]`